Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{9p^3 - 108p^2 + 180p}{-10p^3 + 10p^2 + 20p}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {9p(p^2 - 12p + 20)} {-10p(p^2 - p - 2)} $ $ r = -\dfrac{9p}{10p} \cdot \dfrac{p^2 - 12p + 20}{p^2 - p - 2} $ Simplify: $ r = - \dfrac{9}{10} \cdot \dfrac{p^2 - 12p + 20}{p^2 - p - 2}$ Since we are dividing by $p$ , we must remember that $p \neq 0$ Next factor the numerator and denominator. $ r = - \dfrac{9}{10} \cdot \dfrac{(p - 2)(p - 10)}{(p - 2)(p + 1)}$ Assuming $p \neq 2$ , we can cancel the $p - 2$ $ r = - \dfrac{9}{10} \cdot \dfrac{p - 10}{p + 1}$ Therefore: $ r = \dfrac{ -9(p - 10)}{ 10(p + 1)}$, $p \neq 2$, $p \neq 0$